The thing to blame if you think people don't like it when women are in charge is "role congruity." It’s the theory that the most primitive parts of our brain continue to dictate our ideas about gender roles, with the man setting the rules of cave-house and forging the path to the mammoth den while the woman sticks to just nurturing things. Thus, when a woman steps into a leadership role, it’s supposed to trigger a “gender-role violation” that, consciously or not, stirs us to see her as less capable.

This phenomenon has been supported by research. One study, in 1990, found that female leaders were given more negative and less positive feedback than male leaders, even though they offered the same suggestions and arguments.

In 2003, a group of business-school students were divided into two groups: Half were told that a fictional entrepreneur’s name was Heidi; the other half that it was Howard. Though the students said that Heidi and Howard were both competent and worthy of respect, “Heidi was seen as selfish and not the type of person you would want to hire or work for."

But that was two decades ago. Since then, women have made some, but not a lot, of strides as they scuttle across the Minefield of Success and Likability.

As my former colleague Eleanor Barkhorn pointed out, in 2011, a study of 60,000 people published in Human Relations found that “people who actually had female managers did not give them lower ratings than people who had male managers.”

The rub? “Of the 46 percent of respondents who expressed a preference for their boss's gender, 72 percent said they wanted a male manager.”

In other words, people viewed the female bosses like Chipotle customers view a mildly botched burrito order—fine in the end, just not exactly what they wanted. Not exactly the story of the triumph of feminism.

To help sort through some of these conflicting findings, researchers from Florida International University and the University of North Carolina at Charlotte recently conducted a meta-analysis, published in the Journal of Applied Psychology, of 99 studies that examined perceptions of female leaders.

They found that while male leaders were more likely to rate themselves as more effective than the female leaders were, other people were actually more likely to rate the female leaders as more effective.

When the researchers combined the effect sizes for the self-ratings and the ratings of others, the gender difference in effectiveness evened out to approximately zero.

This backs the recent Atlantic magazine article by Katty Kay and Claire Shipman, which documented how women consistently underestimate their skills and potential, while men tend to overstate them. Meanwhile, men are still paid more for doing the same work and are much better represented in c-suites and in Congress.

"These findings are surprising given that men on average continue to be paid more and advance into higher managerial levels than women," one of the current study authors, Samantha C. Paustian-Underdahl, said in a statement. "Future research needs to examine why women are seen as equally (or more) effective leaders than men, yet are not being rewarded in the same ways."

The good news here is that Paustian-Underdahl's analysis suggests attitudes toward female leaders are changing—there was a tendency for the later studies examined to have a rosier view of the female leaders, but the effect wasn't significant. It also hints that the leadership needs of organizations are shifting, too. Now that so many of us are white-collar pixel-pushers working across cultures and time-zones, there’s less of a need for commandeering foremen and more of a need for open, collaborative, “feminine” bosses gently nudging us to greatness.

As another meta-analysis from 2011 put it, “Leadership now, more than in the past, appears to incorporate more feminine relational qualities, such as sensitivity, warmth, and understanding.”

But this isn’t a total slam-dunk for ambitious women. It seems that what really matters for female leaders is not whether they are “successful,” but how they behave in their role. And when they behave more like men, female leaders haven’t made nearly as much progress as this study might imply.

As the Clayman Institute sociologist and Lean In researcher Marianne Cooperpoints out in the Harvard Business Review, “Scientific research also tells us that male and female leaders are liked equally when behaving participatively (i.e. including subordinates in decision making). But when acting authoritatively, women leaders are disliked much more than men. To be clear, it is not that women are always disliked more than men when they are successful, but that they are often penalized when they behave in ways that violate gender stereotypes.”

And indeed, Paustian-Underdahl and her co-authors also found that the type of workplace made a difference: Men were seen as more effective in male-dominated settings, such as government work, while women were viewed as more capable within female-dominated industries, like education.

The current study also measured “effectiveness,” not likability, and the researchers defined “effectiveness” as any one of the following: “performance or leadership ability; ratings of satisfaction with leaders or satisfaction with leaders’ performance; coding or counting of effective leadership behaviors; or measures of organizational productivity or group performance.”

You could be satisfied that a leader has fostered organizational productivity, while still believing that she’s a shrill bitch.

So while authoritative female leaders might still be far from being well-liked, it’s a pretty big deal to know that female bosses in general are viewed as just as effective—and sometimes more so. It should at least be enough to refute any latent sexist claims that we’re better suited to tending the cave fire. And it should definitely help those self-effacing female leaders stand a little taller.

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